Null Cones and Einstein's Equations in Minkowski Spacetime

TL;DR

This paper explores how Einstein's equations can be consistent with Minkowski spacetime by ensuring the effective metric's null cone respects the background metric, using a generalized eigenvector formalism and gauge freedom.

Contribution

It introduces a generalized eigenvector approach to relate null cones in Minkowski spacetime and shows how gauge freedom can enforce causality constraints.

Findings

Effective curved metric respects flat null cone

Reduced configuration space enforces causality

Gauge transformations form a groupoid, not a group

Abstract

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segr\'{e} classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity.